In triangle ABC, angle B is a right angle, AC = 20, and A = 34. Point D lies on AC so that BD is perpendicular to AC. What is the length of BD?

SAT Right triangles and trigonometry Practice Question 3 (Hard)

Question bank / Right triangles and trigonometry

Math · Geometry and TrigonometryHard

In triangle $ABC$ , angle $B$ is a right angle, $AC = 20$ , and $\tan A = \frac{3}{4}$ . Point $D$ lies on $\overline{AC}$ so that $\overline{BD}$ is perpendicular to $\overline{AC}$ . What is the length of $\overline{BD}$ ?

$7.2$

$12$

$12.8$

$9.6$

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Answer: D

Since $\tan A = \frac{BC}{AB} = \frac{3}{4}$ , the sides are in the ratio $3:4:5$ , so with hypotenuse $AC = 20$ , $BC = 12$ and $AB = 16$ . The area of the triangle is $\frac{1}{2}(16)(12) = 96$ . Using $\overline{AC}$ as the base, $\frac{1}{2}(20)(BD) = 96$ , so $BD = 9.6$ .

Choice A is the length of $\overline{CD}$ (since $CD = BC \cos C = 12 \cdot \frac{3}{5}$ ), not the altitude.

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